Deformations of smooth projective hypersurfaces and the Jacobian ring

Question

It is a well-known result of Griffiths that the pieces of Hodge filtration of a smooth hypersurface $X:= (f=0)$ of degree $d$ in $\mathbb{P}^{n}$ are isomorphic to graded pieces of the Jacobian ring associated to $X$ i.e the ring $R= \mathbb{C}[x_{1},...x_{n}]/J$ where $J$ is the ideal generated by the partial derivatives of $f$ . The isomorphism is given by means of the so-called Poincare residue. There is another isomorphism $R^{d} \cong H^{1}(X,T_{X})$ (both are isomorphic to the space of infinitesimal deformaions of $X$) . I don't know how this isomorphism is explicitly given. In other words, I am wondering that in a concrete example how one can explicitly find the image of an element of $H^{1}(X,T_{X})$ in $R^{d}$under this last isomorphism. More exactly if I have a 1-Cech cocycle of the bundle $T_{X}$, how can I explicitly associate a homogenous polynomial of degree $d$ in $R^{d}$ to it? For simplicity you can consider the family $y^{4}= x(x-1)(x+1)(x-\lambda)$ of plane curves of degree $4$.

Answer 1

For $d\geq 5$, for a smooth plane curve of degree $d$, it is simply not true that the natural map $R_d \to H^1(X,T_X)$ is surjective: there are plenty of abstract deformations which are not plane curves. I am including below the proof that the natural map is surjective when $n\geq 4$, when $n=3$ and $d\leq 3$, and when $n=2$ and $d\leq 4$.

The basic idea is to look at the short exact sequence of sheaves $$ 0 \to T_X \to T_{\mathbb{P}^n}|_X \to \mathcal{O}_{\mathbb{P}^n}(d)|_X \to 0.$$ Using the long exact sequence of cohomology, once you prove that $h^1(X,T_{\mathbb{P}^n}|_X)$ equals $0$, then it follows that the connecting map $$H^0(X,\mathcal{O}_{\mathbb{P}^n}(d)|_X) \to H^1(X,T_X)$$ is surjective. Of course the image of $H^0(X,\mathcal{O}_{\mathbb{P}^n}(d)|_X)$ is $R_d$. Finally, using the long exact sequence of cohomology associated to the Euler sequence, $$ 0 \to \mathcal{O}_{\mathbb{P}^n}|_X \to \mathcal{O}_{\mathbb{P}^n}(1)^{\oplus n+1}|_X \to T_{\mathbb{P}^n}|_X \to 0,$$ to prove that $h^1(X,T_{\mathbb{P}^n}|_X)$ equals $0$, it suffices to prove that $h^1(X,\mathcal{O}_{\mathbb{P}^n}(1)|_X)$ and $h^2(X,\mathcal{O}_X)$ both equal $0$. These cohomologies can be computed from the usual computation of $H^q(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n}(r))$ using the long exact sequence of cohomology associated to the short exact sequence, $$ 0 \to \mathcal{O}_{\mathbb{P}^n}(r-d) \to \mathcal{O}_{\mathbb{P}^n}(r) \to \mathcal{O}_{\mathbb{P}^n}(r)|_X \to 0 $$ For $n\geq 3$, always $h^1(X,\mathcal{O}_{\mathbb{P}^n}(1)|_X)$ equals $0$, and the second vanishing holds so long as $\text{dim}(X) \geq 3$, i.e., $n\geq 4$. It is not hard to prove that the second vanishing also holds if $n=3$ and $d\leq 3$. It definitely fails if $n=3$ and $d=4$, i.e., it fails for quartic K3 surface (although, in fact, the natural map is surjective again for $d\geq 5$).

This brings us to the case that $n=2$, i.e., $X$ is a plane curve. By adjunction and Serre duality on $X$, $h^1(X,\mathcal{O}_X(d))$ equals $0$. So the natural map is surjective if and only if $h^1(X,T_{\mathbb{P}^n}|_X)$ equals $0$. By the Euler sequence again, this holds if and only if the map $$ H^1(X,\mathcal{O}_X) \to H^1(X,\mathcal{O}_X(1))^{\oplus 3} $$ is surjective. If $d$ equals $1$, $2$ or $3$, the second group is already zero. Finally, if $d$ equals $4$, your case, then by Serre duality the transpose map is $$ H^0(X,\mathcal{O}_X)^{\oplus 3} \to H^0(X,\mathcal{O}_X(1)) $$ So you are reduced to the question of whether the restriction map '$$H^0(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}(1)) \to H^0(X,\mathcal{O}_X(1)) $$' is surjective, i.e., whether the next term $H^1(\mathbb{P}^2,\mathcal{O}(-3))$ equals $0$, which it does.

Once again, for $d\geq 5$, the natural map is not surjective.

Regarding your original question, how to "transform" a cocycle into an element of $R_d$: on the basic open affine covers of $\mathbb{P}^2$, write the corresponding Cech 1-cocycle for $T_{\mathbb{P}^2}|_X$ as the coboundary of Cech 0-cochain, which you can do since the cohomology group vanishes. Then take the image 0-cochain in $\mathcal{O}(d)|_X$. The coboundary of this 0-cochain will be the image of your original 0-cochain in $\mathcal{O}_X(d)$, which is zero since the composite map $T_X \to \mathcal{O}_X(d)$ is zero. So your 0-cochain is a 0-cocycle, i.e., a global section in $R_d$.